What is the extreme f(x,y)=x^2+xy+y^2-16y+85 ?

The extreme f(x,y)=x^2+xy+y^2-16y+85 is Minimum(-16/3 , 32/3)

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extreme f(x,y)=x^2+xy+y^2-16y+85

extreme points

f (x, y) = x^{2} + x y + y^{2} - 16 y + 85

Minimum (- \frac{16}{3}, \frac{32}{3})

Solution steps

Find the critical points:

(- \frac{16}{3}, \frac{32}{3})

\frac{\partial ^{2} f}{\partial x ^{2}} = 2

D (x, y) = 3

Check the sign of at each critical point

Check critical point

(- \frac{16}{3}, \frac{32}{3}) :

Minimum

Minimum (- \frac{16}{3}, \frac{32}{3})

f (x) = 4 - 5 x + x^{2}

f (x, y) = x^{2} + x y + y^{2} - 19 y + 120

f (x) = 5 - 4 x - x^{3}

f (x) = x^{3} + 6 x^{2} - 5, - 5 \leq x \leq 2

f (x) = x^{3} - 6 x^{2} + 12 x + 2

What is the extreme f(x,y)=x^2+xy+y^2-16y+85 ?
The extreme f(x,y)=x^2+xy+y^2-16y+85 is Minimum(-16/3 , 32/3)